A family of Eta Quotients and an Extension of the Ramanujan-Mordell Theorem

Abstract

Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (\Gamma_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.